Control of Adaptive Optics

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Control of Adaptive Optics
Don Gavel Director of the Laboratory for Adaptive
Optics UCO/Lick Observatory UC Santa Cruz
ASTRO 289C February 26, 2008
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In this class you will learn

• What is control?
• The concept and architecture of closed loop
  feedback control
• About a basic tool the Laplace transform
• Using the Laplace transform to characterize the
  time and frequency domain behavior of a system
• Manipulating Transfer functions in the system
  analysis
• How to predict statistical performance of the
  controller, including
• Calculating how much atmospheric turbulence is
  rejected and how much remains in the controlled
  wavefront
• Calculating the degree to which measurement noise
  corrupts the controlled wavefront
• How to optimize the controller for overall noise
  performance

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Adaptive Optics Control - the basic idea
Aberrated wavefront phase surfaces
Pupil function
Wavefront correction element (Deformable mirror)
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Feedback control architecture
Deformable mirror
Imaging detector
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Closed loop control
w(t)

e(t)
residual
disturbance
-
gH(f)
y(t)
C(f)
where g loop gain
correction
H(f) Hartmann sensor, sampling dynamics,
computational delay, dynamics of the DM and
drive electronics.
Our goal will be to suppress e(t) (residual) so
that y(t)w(t)
We can design a filter, C(f), into the feedback
loop to

• Stabilize the feedback (i.e. keep it from
  oscillating)
• Optimize performance

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The Laplace Transform Pair

• Example 1 decaying exponential

Transform
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The Laplace Transform Pair
Example 1 (continued), decaying exponential
Inverse Transform
The above integration makes use of the Cauchy
Principal Value Theorem
If F(s) is analytic then
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The Laplace Transform Pair
Example 2 Damped sinusoid
t
0
Im(s)
w
x
-s
Re(s)
-w
x
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Laplace Transform Pairs
H(s)
h(t)
(like lim s0 e-st )
1
unit step
delayed step
0
t
T
unit pulse
0
t
T
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Laplace Transform Properties (1)
Linearity
Time-shift
Dirac delta function transform (sifting
property)
Convolution
Impulse response
Frequency response
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Laplace Transform Properties (2)
System Block Diagrams
Y(s)
y(t)
X(s)
x(t)
H(s)
h(t)
product of input spectrum X(s) with frequency
response H(s) (H(s) in this role is called the
transfer function)
convolution of input x(t) with impulse response
h(t)
Power or Energy
Parsevals Theorem
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Closed loop control (simple example, H(s)1)
W(s)

E(s)
residual
disturbance
-
Y(s)
gC(s)
correction
where g loop gain
Our goal will be to suppress X(s) (residual) by
high-gain feedback so that Y(s)W(s)
solving for E(s),
Note for consistency around the loop, the
units of the gain g must be the inverse of the
units of C(s).
Q1. What would happen if gC(iw) -1 at some w?
What is a good choice for C(s)?...
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The integrator, one choice for C(s)
A system whos impulse response is the unit step
acts as an integrator to the input signal
that is, C(s) integrates the past history of
inputs, x(t)
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The Integrator (2)
In Laplace terminology
An integrator has high gain at low frequencies,
low gain at high frequencies.
Write the input/output transfer function for an
integrator in closed loop
The closed loop transfer function with the
integrator in the feedback loop is
closed loop transfer function
output (e.g. residual wavefront to science camera)
input disturbance (e.g. atmospheric wavefront)
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The integrator in closed loop (2)
where
HCL(s), viewed as a sinusoidal response filter
DC response 0 (Type-0 behavior)
High-pass behavior
and the break frequency (transition from low
freq to high freq behavior) is around w g
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The integrator in closed loop (3)
At the break frequency, w g, the power
rejection is
Hence the break frequency is often called the
half-power frequency
1
1/2
g
w
w2
(log-log scale)
Note also that the loop gain, g, is the bandwidth
of the controller. Frequencies below g are
rejected, frequencies above g are passed. By
convention, g is thus known as the gain-bandwidth
product.
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Now lets apply these concepts to the adaptive
optics controller
measurement noise
residual wavefront
atmospheric wavefront
D actuator voltages
Wavefront sensor

-
DM correction
actuator voltages
Integrator
Deformable Mirror
Discretized model

-
g/s
Note sloppy misuse of notation in the block
diagram, mixing time domain signals and Laplace
domain system transfer functions - this is
common live with it. But dont do it in
equations. (g/s means integrate from 0 to t and
multiply by g)
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Dynamics of each component of the adaptive optics
system
Wavefront sensor transfer function, W(w)
Stare Dt
Computer delay tc
Sample and Hold Dt
Deformable mirror transfer function, D(w)
http//www.okotech.com/publications/sid.pdf
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Optimizing the accuracy of the controller
There are two sources for residual wavefront 1)
atmosphere (aij) 2) measurement noise (nij)
Well use a simple model ignoring the dynamics of
the wavefront sensor and deformable mirror W(s)
1 and D(s) 1
Block diagram

-
g/s
Laplace Domain Equation
residual wavefront
atmospheric wavefront
measurement noise
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Optimizing the accuracy of the controller
We can now solve for a control gain that
optimizes the accuracy of the controller by
trading off atmosphere rejection against
sensitivity to noise.
First, calculate the variance of the residual
atmospheric wavefront without noise
convolution property
Time-average, and, formally, statistical average
Parsevals theorem, and
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Optimizing the accuracy of the controller (2)
Calculating the variance of the residual
atmospheric wavefront without noise
Example, substitute power spectrum of Kolmogorov
turbulence
const depends on wind velocity and turbulence
strength
w
g
HCL(s)2
A(iw)2Dwconst x w-8/3
(log-log scale)
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Optimizing the accuracy of the controller (3)
Calculating the variance of the residual
atmospheric wavefront without noise
change of variables w w / g
By definition of Greenwood frequency, sBW,has
units of radians of wavefront phase.
where
wg is a characteristic bandwidth of the
atmosphere (Greenwoods frequency). g wc is the
gain-bandwidth product of the controller.
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Homework problem 1

• For a disturbance wavefront with a Greenwood
  frequency of 50 Hz, and zero measurement noise,
  calculate the residual closed loop error in an AO
  system in the case where closed loop
  gain-bandwidth product equals
• 10 Hz
• 50 Hz
• 100 Hz
• Calculate this error in nanometers, rms,
  assuming that the Greenwood frequency is
  specified for l 0.5 microns wavelength.

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Optimizing the accuracy of the controller (4)
Now calculate the contribution to the variance of
the residual wavefront due to the measurement
noise
Noise transfer function
residual wavefront due to meas. noise
Noise
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Optimizing the accuracy of the controller (5)
Calculating the contribution to the variance of
the residual wavefront due to the measurement
noise
Use Parsevals theorem to calculate the power
assuming white measurement noise
Note snoise, has units of radians of wavefront
phase. sN has units of radians of phase per
square-root Hz.
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Optimizing the accuracy of the controller (6)
Summary of contributors to residual wavefront
variance
Due to atmosphere
Due to measurement noise
where
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Homework problem 2

• For white measurement noise of 20 nm/vHz (sN
  0.1), and zero wavefront disturbance, calculate
  the residual wavefront error in closed loop in
  the case where closed loop gain-bandwidth product
  equals
• 10 Hz
• 50 Hz
• 100 Hz
• Calculate this error in nanometers, rms.

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Optimizing the accuracy of the controller (7)
The loop gain ( controller bandwidth) can now be
optimized for minimum residual wavefront variance
Optimal gain setting g is

• increased when atmospheric bandwidth (wg)
  increases
• decreased when measurement noise increases

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Homework problem 3

• For the situation in the previous 2 problems (50
  Hz Greenwood frequency and 20 nm/vHz rms
  measurement error), determine the optimum
  gain-bandwidth product for a control loop design.
  Express the answer in Hertz.
• Also compute the total rms residual wavefront
  error, in nanometers, at this gain and verify
  that this optimum is reasonable by comparing to
  the root-sum-square of error terms from each of
  the case studies in problems 1 and 2.

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In conclusion, you have learned

• The architecture and problems associated with
  feedback control systems
• The use of the Laplace transform to help
  characterize closed loop behavior
• How to predict the performance of the adaptive
  optics under various conditions of atmospheric
  seeing and measurement signal-to-noise
• How to optimize the controller for overall noise
  performance, if we know or measure the second
  order statistics of the atmospheric turbulence
  and the measurement noise

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Quick Review
System Block Diagrams
Y(s)
y(t)
X(s)
x(t)
H(s)
h(t)
product of input spectrum X(s) with frequency
response H(s) (H(s) in this role is called the
transfer function)
convolution of input x(t) with impulse response
h(t)
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Stable input-output property system response
consists of a series of decaying exponentials
Transform Domain
Time Domain
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What Instability Looks Like
Transform Domain
Time Domain
Im(s)
x
Re(s)
-s
t
0
Im(s)
w
x
-s
Re(s)
t
-w
x
0
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Control Loop Arithmetic
Y(s)
W(s)

A(s)
output
input
-
B(s)
solving for Y(s),
Instability if any of the roots of the polynomial
1A(s)B(s) are located in the right-half of the
s-plane